Period Count for Fibonacci recurrences modulo primes

Let $p$ be prime, and let $\#(p)$ count distinct periods modulo $p$ of the Fibonacci recurrence and its parity transform over all initial conditions $(a_0,a_1)\in(\mathbb Z/p\mathbb Z)^2$. Let $\alpha$ be the positive integer governing the Pisano period. Period Count for Fibonacci Recurrences modulo $p$. If $p\equiv2,3\pmod5$, then $\pi_A(p)=2(p+1)/\alpha$ for odd $\alpha$ and

$$\#_A(p)=\frac\alpha2(p-1)+1.$$

If $p\equiv1,4\pmod5$, then $\pi_B(p)=(p-1)/\alpha$, with subclasses

$$\#_{B1}(p)=\alpha(p+1)+1,$$

for the two-length case, and

$$\#_{B2}(p)=\alpha(p+2)+1,$$

for the three-length case. In subclass B2, the lengths are $0$, $\pi_B(p)/2$, and $\pi_B(p)$, with multiplicities $2\alpha$ and $p\alpha$ for the latter two. All primes congruent to $11,19\pmod{20}$ belong to B2, while primes congruent to $1,9\pmod{20}$ may belong to either subclass. These proposed classifications and counts are not resolved in the supplied text.